The natural and more promising approach for circumventing this barrier is to somehow ensure that there are no \(\le k\) -cycles in the original graph \(G\) .
Then, any effective zigzag must be long, and even the natural two-layered construction would give us a lower bound of \(\Omega (k)\) .
Indeed, the distance for a pair \(b,c\)  would be \(=2\)  if the pair is in a triangle, versus \(\ge k-1\)  otherwise.
This would be reminiscent of the use of the girth conjecture in lower bounds for multiplicative spanners [1]}, [2]},
whereas the aforementioned graph-products technique is reminiscent of lower bounds for additive spanners [3]}, [4]}, [5]}.The Girth Conjecture and the techniques for additive spanners were already used, of course, for lower bounds against distance oracles as well. However, such lower bounds (and any information-theoretic arguments) cannot prove lower bounds higher than \(m\) ; rather, they are interesting for understanding how much dense graphs can be compressed. Thus, the similarity can only be in spirit.
All we have to do is to prove this gap amplification result for Triangle, amplifying the no-case from triangle-free to \(k^{\prime }\) -cycle free, for all \(4\le k^{\prime } \le k\)  (without unintentionally removing a triangle in the yes-case).
This boils down to the following natural question.
